Population Ecology: First Principles (Second Edition) - Softcover

Vandermeer, John H.; Goldberg, Deborah; Goldberg, Deborah E.

Softcover

ISBN 10: 0691160317 ISBN 13: 9780691160313

Publisher: Princeton University Press, 2013

Synopsis

Ecology is capturing the popular imagination like never before, with issues such as climate change, species extinctions, and habitat destruction becoming ever more prominent. At the same time, the science of ecology has advanced dramatically, growing in mathematical and theoretical sophistication. Here, two leading experts present the fundamental quantitative principles of ecology in an accessible yet rigorous way, introducing students to the most basic of all ecological subjects, the structure and dynamics of populations. John Vandermeer and Deborah Goldberg show that populations are more than simply collections of individuals. Complex variables such as distribution and territory for expanding groups come into play when mathematical models are applied. Vandermeer and Goldberg build these models from the ground up, from first principles, using a broad range of empirical examples, from animals and viruses to plants and humans. They address a host of exciting topics along the way, including age-structured populations, spatially distributed populations, and metapopulations. This second edition of Population Ecology is fully updated and expanded, with additional exercises in virtually every chapter, making it the most up-to-date and comprehensive textbook of its kind. * Provides an accessible mathematical foundation for the latest advances in ecology * Features numerous exercises and examples throughout * Introduces students to the key literature in the field * The essential textbook for advanced undergraduates and graduate students * An online illustration package is available to professors

About this title

About the Author

John H. Vandermeer is the Asa Gray Distinguished University Professor of Ecology and Evolutionary Biology at the University of Michigan. Deborah E. Goldberg is the Elzada U. Clover Collegiate Professor of Ecology and Evolutionary Biology at the University of Michigan.

Excerpt. © Reprinted by permission. All rights reserved.

Population Ecology

FIRST PRINCIPLES

By JOHN H. VANDERMEER, DEBORAH E. GOLDBERG

PRINCETON UNIVERSITY PRESS

Copyright © 2013 Princeton University Press
All rights reserved.

ISBN: 978-0-691-16031-3

Contents

List of Figures............................................................
xi
List of Tables.............................................................
xvii
Preface....................................................................
xix
ONE Elementary Population dynamics.........................................
1
TWO Projection matrices: structured models.................................
30
THREE applications of simple Population models.............................
62
FOUR a closer look at the "dynamics" in Population dynamics................
81
FIVE Patterns and dynamics in space........................................
126
SIX Predator–Prey (consumer–resource) interactions.........................
152
SEVEN disease Ecology......................................................
187
EIGHT competition..........................................................
198
NINE Facilitation and mutualism............................................
225
TEN What this book Was about...............................................
239
Glossary...................................................................
243
References.................................................................
247
Index......................................................................
255

CHAPTER 1

Elementary Population Dynamics

In 1960 the famous cyberneticist Heinz von Foerster and colleagues devisedan equation predicting that the human population would become effectivelyinfinite on Friday the 13th of November, 2026, meaning that at that pointin time all humans would perish because the next individual to be born wouldcrush everyone else—mass death due to squashation! In fact von Foerster andhis colleagues were making a tongue-in-cheek argument to call attention to anissue they thought quite important. This was one, perhaps humorous, exampleof the application of simple quantitative principles of population dynamicsto problems considered important.

Indeed there are many contexts in which it is important to understand thequantitative characteristics of single populations of organisms. In fisheries management,for example, the manager is interested in being able to predict thedensity of a fish population in the future under different management plans. Anagronomist may wish to know the yield of a population of maize plants whenplanted at a particular density; an epidemiologist will want to know the densityof disease-infected humans next month. Many other examples could be citedwith clear practical importance. Of perhaps even more importance are theoreticalapplications that give us a more detailed understanding of more complexecological systems. We might be interested in knowing the rate at which a populationchanges its density in response to selection pressure as part of a generalprogram of understanding the consequences of natural selection under somehypothetical or real constraints. These topics, both applied and theoretical, aretypical of the field called population ecology, and they all start with some basicideas of what single populations of organisms do.

The unit of analysis is, not surprisingly, the "population," a concept that isat once simple and complicated. The simple idea is that a population is simplya collection of individuals. But, as most ecologists intuitively know, the idea ofa population is considerably more complex when one deals with it in any ofthe applied or theoretical contexts alluded to above. To know what size limitsone should place on a fish species one must know not only the number of fishin the population but the size distribution of that population and how thatdistribution relates to the population's reproductive effort. To decide whento take action on the emergence of pest species in an agricultural or forestrycontext, the distribution of individuals in various life stages must be known.In deciding whether a species is threatened with extinction, its distribution inspace and movement among subpopulations (i.e., metapopulation dynamics)is far more important than simply its numerical abundance. And, to use themost frequently cited example, given the huge variation in the consumptionof resources per person around the globe, the absolute numbers of the humanpopulation may be much less important than the activities undertaken by themembers of that population—doomsday could be at hand well before 2026.

Thus the subject of population ecology can be a very complicated oneindeed. But, as in the case of any science, we begin by assuming that it israther simple. We eliminate the complications, make simplifying assumptions,and try, as much as possible, to develop general principles that might forma skeleton onto which the flesh of real-world complications might meaningfullybe attached. In 2013 it seems that, unlike some other fields of biology,population ecology has a certain core subject matter that has come to be the"conventional wisdom." That is, wherever a university course in populationecology is taught, pretty much the same material is covered, at least at thebeginning of the course. This text is our attempt to present that core materialprecisely, and this chapter covers the first two essential ideas—the densityindependence and density dependence of population growth.

Density Independence: The Exponential Equation

It is often surprising how quickly a self-reproducing event becomes a bigevent. The classic story is this: suppose you have a pond with some lily padsin it, and suppose each lily pad replicates itself once per week. If it takes ayear for half the pond to become covered with lily pads, how long will it takefor the entire pond to become covered? If one does not think too long or toodeeply about the question, the quick answer seems to be about another year.But with a moment's reflection one can retrieve the correct answer—only onemore week.

This simple example has many parallels in real-world ecosystems. A pestbuilding up in a field may not seem to be a problem until it is too late. A diseasemay seem much less problematical than it really is. The simple problemof computing the "action threshold" (the population density a pest mustreach before you have to spray pesticide) requires the ability to predict howlarge a population will be based on its prior behavior. If half the plants in thefield are attacked within three months, how long will it be before they are allattacked?

To understand even the extremely simple example of the lily pads, oneconstructs a mathematical model, usually quite informally, in one's head. Ifall the lily pads on a pond replicate themselves once per week, in a pond half-filledwith lily pads each one of those lily pads will replicate itself in the nextweek and thus the pond will be completely filled up. To make the solutionto the problem general, we simply say the same thing, but instead of labelingthe entities lily pads, we call them something general, say organisms. Iforganisms replicate themselves once per week, by the time the environmentis half full, it will take only one week to become completely full. Implicitly,the person who makes such a statement is simply stating verbally the followingequation:

N_t+1 = 2N_t. (1)

N is the number of organisms, in this case lily pads. Instead of N, say lilypads, and instead of t, say this week, and instead of t + 1, say next week, andequation 1 expresses simply, "The number of lily pads next week will be equalto twice the number this week."

Writing down equation 1 is no different than making any of the statementsthat were made previously about it. But by making it explicitly a mathematicalexpression, we bring to our potential use all the machinery of formalmathematics. And that is actually good, even though beginning studentssometimes don't think so.

Using equation 1 we can develop a series of numbers that reflect thechanges of population numbers over time. For example, consider a populationof herbaceous insects. If each individual produces a single offspringonce per week, and if those offspring mature and each produces an offspringwithin a week, we can apply equation 1 to see exactly how many individualswill be in the population at any point in time. Beginning with a single individual,we have, in subsequent weeks, 2, 4, 8, 16, 32, 64, 128, and so on. Ifwe change the conditions such that the species replicates itself twice per week,equation 1 becomes

N_t+1 = 3N_t (2)

(with a 3 instead of a 2, because before we had the individual and the singleoffspring it produced but now we have the individual and the two offspringit produced). Now, beginning with a single individual, we have, in subsequentweeks, 3, 9, 27, 81, 243, and so on.

We can use this model in a more general sense to describe the growth of apopulation for any rate of production of offspring at all (not just 2 and 3 asabove). That is, write

N_t+1 = λN_t, (3)

where λ can take on any value at all. λ is frequently called the finite rate ofpopulation growth (or the discrete rate).

It may have escaped notice in the above examples, but either of the seriesof numbers could have been written with a much simpler mathematical notation.For example, the series 2, 4, 8, 16, 32, is a actually 2¹, 2², 2³, 2⁴, 2⁵, whilethe series 3, 9, 27, 81, 243 is actually 3¹, 3², 3³, 3⁴, 3⁵. So we could write

N_t = λ^t, (4)

which is just another way of representing the facts described by equation 3.(Remember, we began with a single individual, so N₀ = 1.0.)

For further exposition we wish to express the constant λ in a differentfashion. It is a general rule that any number can be written in a large numberof ways. For example, the number 4 could be written as 8/2 or 9 - 5 or2² or in many other ways. In a similar vein, an abstract number, say λ, couldbe written in any number of ways: λ = 2b, where b is equal to λ/2, or λ = 2^b,in which case b = ln(λ)/ln(2) (where ln stands for natural logarithm). Forreasons that will be obvious to the reader not too far removed from the elementarycalculus class, if we represent λ as 2.7183^r, a powerful set of mathematicaltools immediately becomes available. The number 2.7183 is Euler'sconstant, usually symbolized as e (actually, 2.7183 is rounded off and thus isonly approximate). It has the important mathematical property that its naturallogarithm is equal to 1.0 or ln(e) = 1.

So we can rewrite equation 4 as

N_t = e^rt, (5)

which is the classical form of the exponential equation (where λ has beenreplaced with e^r). One more piece of mathematical manipulation is necessaryto complete the toolbox necessary to model simple population growth.Another seemingly complicated but really rather simple relationship that isalways learned (but frequently forgotten) in elementary calculus is that therate of change of the log of any variable is equal to the derivative of thatvariable divided by the value of the variable. This rule is more compactlystated as

d(ln N)/dt = dN/Ndt. (6)

So take the natural logarithm of equation 5 as

ln(Nt) = rt

and differentiate with respect to t to obtain

d(ln N)/dt = r, (7)

and we can then use equation 6 to substitute for the left-hand side of equation7 to obtain

dN/Ndt = r.

After multiplying both sides by N, we obtain

dN/dt = rN. (8)

Equations 5 and 8 are the basic equations that formally describe an exponentialprocess. Equation 8 is the differentiated form of equation 5, and equation5 is the integrated form of equation 8. They are thus basically the sameequation (and indeed are quite equivalent to the discrete form—equation 3).Depending on the use to which they are to be put, any of the above formsmay be used, and in the ecological literature one finds all of them. Their basicgraphical form is illustrated in figure 1.1.

In the examples of exponential growth introduced above, the parameter rwas introduced as a birth process only. The tacit assumption was made thatthere were no deaths in the population. In fact, all natural populations face thereality of mortality, and the parameter of the exponential equation is really acombination of birth and death rates. More precisely, if b is the birth rate (numberof births per individual per time unit) and d is the death rate (number ofdeaths per individual per time unit), the parameter of the exponential equation is

r = b - d, (9)

where the parameter r is usually referred to as the intrinsic rate of naturalincrease.

One other simplification was incorporated into all of the above examples.We always presumed that the population in question was initiated with a singleindividual, which almost never happens in the real world. But the basicintegrated form of the exponential equation is easily modified to relax thissimplifying assumption. That is,

N_t = N₀e^rt, (10)

which is the most common form of writing the exponential equation. Thusthere are effectively two parameters in the exponential equation, the initialnumber of individuals, N₀, and the intrinsic rate of natural increase, r.

Putting the exponential equation to use requires estimation of these twoparameters. Consider, for example, the data presented in table 1.1. Here wehave a series of observations over a five-week period of the number of aphidson an average corn plant in an imaginary cornfield.

As a first approximate assumption, let us assume that this population originatesfrom an initial cohort that arrived in the field on March 18 (one weekprior to the initial sampling). We can apply equation 10 to these data mosteasily by taking logarithms of both sides, thus obtaining

ln(Nt) = ln(N₀) + rt, (11)

which gives us a linear equation relating the natural logarithm of the numberof aphids to time (where we code March 18 as time = 0, March 25 astime = 1, April 1 as time = 2, etc.). In figure 1.2 is a graph of this equationalong with the original data points, and in figure 1.3 is a graph of the originaldata along with the fitted curve on arithmetic axes.

From these data we have the estimate of 1.547 aphids per parent aphid perweek added to the population (i.e., the intrinsic rate of natural increase, r, is1.547, which is the slope of the line in figure 1.2). The intercept of the regressionis -4.626, which indicates that the initial population was .0098 (i.e., theantilog of -4.626 is .0098), which is an average of about one aphid per 100plants. Now, if we presume that once the plants become infected with morethan 40 aphids per plant the farmer must take some action to try to controlthem, we can use this model to predict when, approximately, this time willarrive. The regression equation is

ln(N of aphids) = -4.626 + 1.547t,

which can be rearranged as

t = [ln(N of aphids) + 4.626] / 1.547.

The natural log of 40 is 3.69, so we have

t = (3.69 + 4.626) / 1.547 = 5.375.

Translating this number into the actual date (remember that April 22 wastime = 5), we see that the critical number will arrive about April 24 (at3:00 p.m. on April 24, theoretically).

In practice, calculations like these exclude many complicating factors, andwe should never take too seriously exact predictions the model makes. On theother hand, April 24 really does represent the best prediction we have basedon available data. It may not be a very good prediction, but it is the only oneavailable. Furthermore, it may seem quite counterintuitive that, having takenfive full weeks to arrive at only 14 aphids per plant, in only two more daysthe critical figure of 40 aphids per plant will be realized—such is the natureof exponential processes. A simple model like this could help the farmer planpest control strategies.

Density Dependence

In the previous section we showed that any population reproducing at a constantper capita rate will grow according to the exponential law. Indeed, thatis the very essence of the exponential law: each individual reproduces at a constantrate. However, the air we breathe and the water we drink are not completelypacked with bacteria or fungi or insects, as they would be if populationsgrew exponentially forever. Something else must happen. That something elseis sometimes referred to as intraspecific competition, which means that the performanceof the individuals in the population depends on how many individualsare in it. More generally, it is referred to as density dependence. It is acomplicated issue that has inspired much debate and acrimony in the past andstill forms an important base for more modern developments in ecology.

The idea was originally associated with the human population, and wasbrought to public attention as early as the nineteenth century by Sir ThomasMalthus (1960 [1830]). It was formulated mathematically by Verhulst (1838)as the "true law of population" (Doubleday 1842); Verhulst's version is betterknown today as the logistic equation (see below). Later, Pearl and Reed(1920), in attempting to project the human population size of the UnitedStates, independently derived the same equation. Associated with its mathematicalformulation was a series of laboratory studies with microorganismsin the first part of the twentieth century. Most notable were those of Gause(1934), in which population growth was studied from the point of view ofcompetition, both intra- and interspecific.

In the early part of the twentieth century a variety of terms were introduced,all of which essentially referred to the same phenomenon of approaching somesort of carrying capacity through a differential response of per capita populationgrowth rate to different densities. Chapman (1928) formulated the idea in termsof "environmental resistance." Howard and Fiske (1911) categorized mortalityfactors as either "catastrophic," in which some proportion of the populationdied regardless of its density, or "facultative," which caused an increasingproportion of the population to die with the increasing density of the population.In 1928 Thompson redefined catastrophic as "general" and "independent"of density and facultative as "individualized" and "dependent" on density, andlater Smith (1935) proposed the density-dependent / density-independent gradient.Thus by the 1930s the dichotomy of density independence versus densitydependence had taken firm root after having been sown not long after the turnof the century.

In the 1930s Nicholson (1933) and his colleague Bailey (Nicholson andBailey 1935) formalized the concept of regulation through density-dependentfactors and clearly associated the idea of intraspecific competition with densitydependence. In Nicholson and Bailey's conceptualization of densitydependence, four points were proposed: (1) population regulation must bedensity dependent; (2) predators and parasites, as well as intraspecific competition,may function as density-dependent forces; (3) more than density dependencealone may function to determine actual population size; and (4) densitydependence does not always function to regulate population density.

(Continues...)

Excerpted from Population Ecology by JOHN H. VANDERMEER, DEBORAH E. GOLDBERG. Copyright © 2013 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: Princeton University Press
• Publication date: 2013
• Language: English
• ISBN 10: 0691160317
• ISBN 13: 9780691160313
• Binding: Paperback
• Edition number: 2
• Number of pages: 288